Linux

and the machine is definitely past its prime, you may have a tough time finding an easy Linux to install on it. BUT, take a look at Slackware, it's been around since almost the beginning, and is well supported. May require a bit of "getting your hands dirty", meaning you'll be doing quite a bit by hand, but once you get it going, it is very stable.
Other alternatives would be maybe something along the lines of Puppy Linux. There are various "flavors" (Debian, 'Buntu, Slackware...I like Slacko Puppy myself) and these can be installed and ran from CD or USB thumbdrive, as a "live distro", to see if they are compatible with your hardware.
Good luck!

Latest Centos7 is Good for Web hosting.

Ubuntu and Fedora are two best options according to your requirements.

As far as hosting a web site almost anything will do it as long as all you have are some basic apache requirements just to say you're putting a page up.

Go much beyond that and that hardware is gonna kill you. At the same time you're probably going to put more time into finding a distro that works well with that hardware than you will getting a website up once you have it running. If you're the guy that likes to bang around on things just to get it running then sure have fun. But from a practical matter you're putting in time that has little to no market value in the real world. No companies are out there hoping to find that one guy that can setup a PIII on linux for them and willing to pay the big bucks to make it happen... and if they exist they're to stupid to want to work for.

Go get some cheap 2010 hardware. It will be easier for you to load on and when you get ready to do something a bit more practical like setup tomcat, jboss, maybe even an easy mysql load behind it to support it you'll have something you can run a mini lab project with.